A particle-in-Burgers model: theory and numerics - Laboratoire de ...
A particle-in-Burgers model: theory and numerics - Laboratoire de ...
A particle-in-Burgers model: theory and numerics - Laboratoire de ...
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Mo<strong>de</strong>l <strong>and</strong> motivation Auxiliary steps Results h = 0: coupl<strong>in</strong>g h = 0: <strong>de</strong>f<strong>in</strong>ition, uniqueness h = 0: <strong>numerics</strong>, existence The coupled problem<br />
Case h = h(t): cont<strong>in</strong>uous <strong>de</strong>pen<strong>de</strong>nce en h(·)<br />
Uniqueness proof for the coupled problem relies on a Gronwall <strong>in</strong>equality,<br />
which <strong>in</strong> turn relies on a Lipschitz <strong>de</strong>pen<strong>de</strong>nce estimate for the map<br />
h(·) ↦→ u(·,·) .<br />
Theorem (<strong>de</strong>pen<strong>de</strong>nce of u on the path h(·))<br />
Assume u, û are entropy solutions correspond<strong>in</strong>g to the <strong>particle</strong>s located at<br />
h(·), ˆ h(·), respectively, with h(0) = 0 = ˆ h(0) <strong>and</strong> same <strong>in</strong>itial datum u0.<br />
Assume û ∈ L ∞ (0, T; BV(R)). Then for a.e. t ∈ (0, T),<br />
t<br />
u(t,·)−û(t,·) L1 (R) C(u∞,û∞,ûBV,λ) |h<br />
0<br />
′ (s)− ˆ h ′ (s)| ds.<br />
Arguments:<br />
– change of variables y = x − h(t), resp. x − h ′ (t). Two eqns, both with<br />
s<strong>in</strong>gularity at zero, come out, with different fluxes of the k<strong>in</strong>d u ↦→ u2<br />
2 − h′ (t)u.<br />
– use the techniques of <strong>de</strong>pen<strong>de</strong>nce of entropy solutions on the flux function<br />
(BV regularity nee<strong>de</strong>d!): Kuznetsov, Bouchut-Perthame, Karlsen-Risebro... :<br />
the C 1 norm of the difference of the fluxes pops up, which yields |h ′ − ˆ h ′ |<br />
– use Lipschitz <strong>de</strong>pen<strong>de</strong>nce of the germ on h ′ to <strong>de</strong>scribe additional (small)<br />
“non-dissipation” term com<strong>in</strong>g from the <strong>in</strong>terface.